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matt grime
#28
Jan10-06, 03:50 AM
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P: 9,398
Quote Quote by Don Aman
Are there Lie algebras without products? Sure, simply don't include a product. Are there Lie algebras which are not isomorphic to algebras with product? No. As far as I can tell, the answer to your question is either "obviously yes" or "obviously no", depending on your view.

which is why i included the words 'naturally occuring', that was sort of the key point in the question that didn't make it vacuous. indeed i gave a formal example of a lie algebra that didn't come with a product

however, as for the example given two tangents at the identity what is [ab] of them? seriously, i'm an algebraist, know nothing of differential geometry, i thought the tangent space was the span of the operators [tex]\partial_x[/tex] which do have a natural product since they are in the associative algebra of differential operators, and that the bracket was ab-ba in this space.

now, if as is entirely possible, there is another definition of tangent space that doesn't require you to evaluate things in this algebra, but which takes in a,b and spits out [ab] naturally then that'd be what the question wanted to know about.